maanantai 15. elokuuta 2011

Christian Wolff: Elements of all mathematical sciences (1710)

The mathematical magnum opus of Christian Wolff is undoubtedly his Der Anfangs-Gründe aller mathemathischen Wissenschaften, which covers in four volumes areas such as arithmetics, astronomy, hydraulics and algebra. The book was apparently heavily used in teaching, because several editions were printed: the one I read hails from 1738. In addition to the book itself, Wolff also wrote a supplement with trigonometric tables, a shortened version and a Latin translation. Apparently he had already realised that the true source of wealth for academics are good text books.

The book is also famous as the target of Hegel's criticism in his Wissenschaft der Logik, where Hegel makes fun of how Wolff tries to formalise mere handicrafts like constructing a window. Indeed, it is rather peculiar to find in the section on architecture definitions of windows and doors (window, for instance, is an opening in the wall from which light can fall into the building). Even more disturbing is the section on artillery, where Wolff defines such things as bombs, hand grenades and mines (e.g. hand grenade is a bomb small enough to be thrown).

As is already apparent, the book would nowadays be read by future engineers and not by future mathematicians. Even in the case of arithmetics and geometry Wolff is more interested of the practical capacities implicit in these sciences: arithmetics, for instance, Wolff says to be of use in counting money. Indeed, the sciences dealt with are as mathematical as the method of Spinoza in Ethics is geometrical. Nowadays we would say that both Spinoza and Wolff applied or at least attempted to apply the so-called axiomatic method. Wolff's book even contains a description of mathematical or axiomatic method, which is undoubtedly the most interesting part of the book for a modern philosopher.

The basic idea of the axiomatic method is that certain propositions are to be developed from other propositions of a more foundational level. Some of these propositions merely tell us what things are or define them, while others or axioms spell out some further characteristics of things. I shall now concentrate on what Wolff has to say about definitions and leave the axioms to my next writing.

It has been customary to separate mere verbal or nominal definitions, which merely explain how certain words are to be used, from real definitions describing the essence of the defined thing. Although the distinction seems clear, it is somewhat difficult to determine when we have truly defined the essence of a thing or even whether thungs have any essences.

Wolff accepts the traditional notion of a nominal definition, but his idea of a real definition seems more creative. Real definition, for Wolff, tells how a thing is generated. Although in one example – that of a solar eclipse – Wolff clearly refers to general causes of a phenomenon, in most cases the generation refers to an actual production of a thing. Thus, Wolff suggests that a real definition of a circle describes how a circle is drawn.

Wolff has replaced the question about essences of things with a more pragmatic aim of extending our abilities to do things. This is clearly in line with Wolff's rather pragmatic attitude towards sciences. Indeed, many of the theorems in Anfangs-Gründe are just such real definitions: we are first told the nominal definition of a window, but only later on do we learn how to build windows. Although German idealists work on a more refined level than at mere handicrafts, one might think that Wolff's idea is a sort of precursor of the later idea of the primacy of practical reason.

According to Wolff, sciences are then more about how to do things than about what things are, that is, sciences are intrinsically connected or even identifiable with technologies. The most interesting consequences of this connection arise in the realm of pure mathematics. Notably, Wolff defines arithmetics not as a science of numerical relationships, but of methods for discovering numbers: 5 + 7 = 12 does not represent a relation between certain objects or sets of objects, but tells that by adding five to seven one gets a new number, that is, twelve.

The difference between the two interpretations of arithmetics is not so obvious when we remain at the level of finite set of numbers, because within a finite set we could at least in principle check whether the set contains a number having a desired property. But in the case of infinite sets the difference becomes more obvious. For instance, it is not at all obvious that we can find solutions to some difficult equations and we cannot even go through all natural numbers to find out if they do have.

This difference actually is what separates the so-called Platonist and constructivist interpretations of mathematics: the former focuses on the mathematical relations and supposes that these relations do exist, even if we cannot know anything about them, while the latter focuses on the actual methods of solving mathematical problems and regards it meaningless to discuss whether these problems would have solutions without any method of solving them. Wolff is unconsciously heading towards the constructivist option, which is interesting, because I think that some of the German idealists have affinities with a constructivist reading of mathematics.

So much for definitions,next time I shall be looking at what Wolff says about axioms, postulates, theorems and problems.